Continuity of Discrete Homomorphisms of Diffeomorphism Groups

Geometry & Topology 19 (2015) 2117–2154 msp

Continuity of discrete homomorphisms of diffeomorphism groups

SEBASTIAN HURTADO

Let M and N be two closed C 1 manifolds and let Diffc.M / denote the group of C 1 diffeomorphisms isotopic to the identity. We prove that any (discrete) group homomorphism between Diffc.M / and Diffc.N / is continuous. We also show that a nontrivial group homomorphism ˆ Diffc.M / Diffc.N / implies that dim.M / W ! Ä dim.N / and give a classiﬁcation of such homomorphisms when dim.M / dim.N /. D 57M99; 55Q33, 55Q32

1 Introduction

Let M be a C 1 manifold and let Diffc.M / be the group of C 1 diffeomorphisms of M which are isotopic to the identity and compactly supported (if M is noncompact). A well-known theorem of Mather, Herman, Thurston and Epstein states that Diffc.M / as a discrete group is simple (see Banyaga[4]).

In[7], Filipckewicz proved the following result: Given two C 1 manifolds M and N , the groups Diffc.M / and Diffc.N / are isomorphic (as discrete groups) if and only if the manifolds M and N are C 1 diffeomorphic. Moreover, if such an isomorphism exists it is given by conjugation with a ﬁxed element of Diff.M /. Filipckewicz’s theorem implies in particular that two different smooth exotic 7–spheres have different diffeomorphism groups. The focus of this paper is the study of abstract group homomorphisms between groups of diffeomorphisms. We will make no assumptions on the continuity of the homomorphisms. Let M and N be two manifolds and let ˆ Diffc.M / Diffc.N / be a W ! group homomorphism. Observe that the simplicity of these groups implies that ˆ is either trivial or injective.

Some examples of homomorphisms of the type ˆ Diffc.M / Diffc.N / are the W ! following. (1) Inclusion If M N is an open submanifold of N , the inclusion gives a homomorphism ˆ Diffc.M / Diffc.N /. W !

Published: 29 July 2015 DOI: 10.2140/gt.2015.19.2117 2118 Sebastian Hurtado

(2) Coverings If p N M is a ﬁnite covering, every diffeomorphism f W ! 2 Diffc.M / can be lifted to a diffeomorphism fz of N . These lifts in many cases give a group homomorphism. For example, this is the case when p is an irregular cover (ie Deck.M=N / Id ), as in this case he lift f of any diffeomorphism f D f g z is unique.

(3) Bundles The unit tangent bundle UT .M /, the Grassmannian Grk .M /, consisting of k –planes in T .M /, product bundles, symmetric products and other bundles over M admit a natural action of Diffc.M /.

Observe that all the homomorphisms described above are in some sense continuous maps from Diffc.M / to Diffc.N / in their usual C 1 topologies. The main result of this paper shows that this is always the case when M is closed. Before stating our main result we need the following deﬁnition:

Deﬁnition 1.1 For any compact subset K M , let DiffK .M / denote the group of dif- Â feomorphisms in Diffc.M / supported in K. A group homomorphism ˆ Diffc.M / W ! Diffc.N / is weakly continuous if for every compact set K M , the restriction Â ˆ of ˆ to DiffK .M / is continuous in the weak topology; see Section 2.3. jDiffK .M / Our main theorem is the following:

Theorem 1.2 Let M and N be C manifolds. If ˆ Diffc.M / Diffc.N / is a 1 W ! discrete group homomorphism, then ˆ is weakly continuous.

It is worth pointing out that there is no assumption on the dimensions of the manifolds. As a consequence of Theorem 1.2, we will obtain a classification of all possible homomorphisms in the case when dim.M / dim.N /. Mann[16] gave a classification of all the nontrivial homomorphisms ˆ Diffc.M / W ! Diffc.N / in the case that N is 1–dimensional. She showed using one-dimensional dynamics techniques (Kopell’s lemma, Szekeres theorem etc) that M must be one- dimensional and that these homomorphisms are what she described as “topologically diagonal”. In the case when M N R, topologically diagonal means that ˆ is given in D D the following way: take a collection of finite disjoint open intervals Ij R and diffeomorphisms fj M Ij . For each diffeomorphism g Diffc.R/ the action W ! 2 1 of ˆ.g/ in the interval Ij is given by conjugation by fj , ie ˆ.g/.x/ figf .x/ D i for every x Ii . The action is defined to be the identity everywhere else. 2 We will prove a generalization of the previous result:

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Theorem 1.3 Let M and N be C manifolds, let N be closed and dim.M / 1 dim.N /. If ˆ Diffc.M / Diffc.N / is a nontrivial homomorphism of groups, then W ! dim.M / dim.N / and ˆ is “extended topologically diagonal”. D The definition of extended topologically diagonal is given in Definition 3.12 and is a generalization of what we previously described as topologically diagonal by allowing the possibility of taking finite coverings of M and embedding them into N ; see Definition 3.12. Observe that as a corollary of Theorem 1.3 we obtain another proof of Filipckewicz’s theorem. The result also gives an affirmative answer to a question of Ghys (see[11]) asking whether a nontrivial homomorphism ˆ Diffc.M / Diffc.N / implies that W ! dim.N / dim.M /. The rigidity results we obtain can be compared to analogous results obtained by Aramayona and Souto[1], who proved similar statements for homomorphisms between mapping class groups of surfaces and to deep results of Margulis about rigidity of lattices in linear groups; see[21, Chapter 5].

1.1 Ingredients and main idea of the proof

The key ingredient in the proof of Theorem 1.2 is a theorem of E Militon. This theorem says roughly that given a sequence hn converging to the identity in Diffc.M / sufficiently fast (a geometric statement), one can construct a finite subset S of Diffc.M /, such that the group generated by S contains the sequence hn and such that the word length lS .hn/ (see Definition 2.2) of each diffeomorphism hn in the alphabet S is bounded by a sequence kn not depending on hn (an algebraic statement); see Theorem 2.3 for the correct statement. Militon’s theorem is a generalization of a result of Avila[3] about Diff .S1/ and is related to results of Calegari and Freedman[6]. Militon’s result is strongly related to the theme of distortion elements in geometry group theory; see Gromov[12, Chapter 3]. An element f of a finitely generated group G n with generating set S is called distorted if limn lS .f /=n 0. !1 D Consider for example the group BS.2; 1/ a; b bab 1 a2 . This group can be D f j D g thought as the group generated by the functions a x x 1 and b x 2x in n W 7! C W 7! Diff.R/. Observe that bnab n a2 for every n and therefore a is distorted (in some D sense a is “exponentially” distorted).

An element f Diffc.M / is said to be recurrent in Diffc.M / if f satisﬁes that n2 1 lim inf dC .f ; Id/ 0, for example an irrational rotation in S is recurrent (see 1 D Section 2.3). A corollary of Militon’s theorem is that any recurrent element in Diffc.M /

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is distorted in some finitely generated subgroup of Diffc.M /. Even more, it implies that such an element is “arbitrarily distorted” in the sense that f could be made as distorted as one wants (see[6] for a precise definition). For a nice exposition of the concept of distortion in transformation groups and the definition of the different types of distortion, see[6]. Observe that distorted elements are preserved under group homomorphisms and therefore if one wants to understand the existence of a group homomorphism, it might be fruitful to understand the distortion elements of the groups involved in the homomorphism. One example of this approach can be seen in the work of Franks and Handel[10]. They proved that any homomorphism from a large class of higher rank lattices (for example SL3.Z/) into Diff.S/ (the group of diffeomorphisms of a surface preserving a measure ) is finite using the fact that has distorted elements and showing that any distorted element f Diff.S/ fixes the support of the measure . For a more 2 concrete statement of this and a clear exposition, see Franks[9]. The main idea of how to use Militon’s theorem to obtain our rigidity results is illustrated in Section 2.2 with a motivating example that encloses the main idea of Theorem 1.2 and is the heart of this paper.

1.2 Outline

The paper is divided as follows. In Section 2, we will use Militon’s theorem to prove Lemma 2.1, which is the main tool to prove Theorem 1.2. In Section 3, we prove Theorem 1.3 assuming that ˆ is weakly continuous; this section is independent of the other sections. In Section 4, we establish some general facts about two constructions in group actions on manifolds that we will use for the proof of Theorem 1.2. In Section 5, we show that ˆ is weakly continuous using the results of Section 2 and the facts discussed in Section 4, ﬁnishing the proof of Theorems 1.2 and 1.3. Finally, in Section 6, we end up with some questions and remarks related to this work. Acknowledgements The author would like to thank the referee, Kathryn Mann, Ben- son Farb and especially Emmanuel Militon for useful comments and corrections on a previous version of this paper. The author is in debt with his advisor, Ian Agol, for his help, encouragement and support. This research was partially supported by NSF grant number DMS-1105738.

2 Main technique

This is the most important section of the chapter: we will prove Lemma 2.1 which is a slightly weaker version of Theorem 1.2. The main ingredient in the proof of Lemma 2.1

Geometry & Topology, Volume 19 (2015) Continuity of homomorphisms of diffeomorphism groups 2121 is Theorem 2.3. We will discuss a motivating example in Section 2.2 which avoids some necessary technicalities in the proof of Lemma 2.1 but contains the central point of our discussion.

2.1 Notation

Throughout this section we make use of the following notation: M and N denote C 1 manifolds, Diffc.M / will be the group of C 1 diffeomorphisms which are compactly supported and isotopic to the identity. The letter ˆ will always be a discrete group homomorphism ˆ Diffc.M / Diffc.N / unless stated otherwise. For a compact W ! set K, we let DiffK .M / be the subgroup of Diffc.M / consisting of diffeomorphisms supported in a compact set K.

We will consider Diffc.M / as a topological space endowed with the weak topology. This topology is metrizable and we will denote by d any metric that induces such topology. For more information about the weak topology see Section 2.3. We are now in position to state the main lemma of this section.

Lemma 2.1 Let M and N be C 1 manifolds, let K be any compact subset of M and let ˆ Diffc.M / Diffc.N / be a discrete group homomorphism. Suppose hn is W ! a sequence in DiffK .M / such that

lim dC .hn; Id/ 0: n 1 D !1 Then ˆ.hn/ contains a subsequence converging to a diffeomorphism H , which is an f g isometry for a C 1 Riemannian metric on N .

Observe that the isometry H in Lemma 2.1 is necessarily homotopic to the identity, and therefore if the manifold N does not admit a metric with isometries homotopic to the identity (as for example is the case when N is a closed surface of genus g 2), the previous theorem implies the continuity of the homomorphism ˆ restricted to DiffK .M /. Before stating Militon’s theorem, we will need the following standard deﬁnition:

Definition 2.2 Let G be a finitely generated group and let S be a finite generating 1 set. We assume that S S . Given an element g G , the word length lS .g/ of g D 2 in s is defined as the minimum integer n such that g can be expressed as a product

g s1s2 sn; D where each si S . In a more technical language, lS .g/ is the distance of g to the 2 identity in the Cayley graph corresponding to S .

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We are now in position to state Militon’s theorem:

Theorem 2.3 Let M be a compact manifold. There exist two sequences n 0 f g ! and kn of positive real numbers such that for any sequence hn of diffeomorphisms in f g Diffc.M / satisfying dC .hn; Id/ n; 1 Ä there exists a ﬁnite set S Diffc.M / such that (1) hn belongs to the subgroup generated by S ,

(2) lS .hn/ kn . Ä Remark 2.4 The previous theorem is also true in the case where M is not compact and Diffc.M / is replaced with DiffK .M / for a compact subset K of M .

Remark 2.5 The set S depends on the sequence hn , but the sequences n and kn f g f g are independent of the choice of hn .

This result is a generalization of a theorem of Avila[3], who proved the same result for 1 the Diffc.S / case. The proof of the theorem is related to a construction of Calegari and Freedman[6], who showed that an irrational rotation of S2 is “arbitrarily distorted” 2 in Diffc.S /. For sake of completeness, we give a rough summary of how Theorem 2.3 is proved in[3] and Militon[18]. The proof of Theorem 2.3 has two steps. The first step is to show that an element in Diffc.M / close to the identity can be written as a product of commutators of elements which are close to the identity; this step is proved using a KAM theory technique. The first step allows one to reduce the general case to the case where all fn are commutators supported in small open balls. The second step consists of using a finite set of diffeomorphisms to encode the sequence fn into a finite S ; f g this step uses some clever algebraic tricks similar to the ones used in the proof of the simplicity of Diffc.M / by Thurston (see[4, page 23]). The tricks depend heavily on the fact that the fn are products of commutators. To give a hint of how powerful Theorem 2.3 can be for our purposes, observe the following. Let hn be a sequence in Diffc.M / supported in a compact set K and converging to the identity in Diffc.M / as in Theorem 2.3. As a consequence of Theorem 2.3, the sequence hn is generated by a finite set of diffeomorphisms S and f g therefore the set ˆ.S/ generates a group containing ˆ.hn/ . As a consequence, there f g exists a compact set K N (namely, the union of the support of the generators) such 0

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that every element of the sequence ˆ.hn/ is supported in K . This statement is not f g 0 easy to deduce by elementary means. The following example is fundamental and contains the main idea in the proof of Lemma 2.1.

2.2 Motivating example

2 2 2 Let S be the 2–sphere and let ˆ Diffc.S / Diffc.S / be a homomorphism of W ! groups. As a consequence of Theorem 1.3 we are going to show that ˆ is induced by conjugation via an element of Diff.S2/. In particular, f and ˆ.f / are conjugate in 2 2 Diff.S / for every f Diffc.S /. In the next proposition we show that this is true for 2 the most basic diffeomorphisms of S2 , namely rotations in SO.3/.

2 2 2 Proposition 2.6 Let S be the 2–sphere and let ˆ Diffc.S / Diffc.S / be a group W ! homomorphism. For any rotation R in SO.3/, we have that C ˆ.R/ preserves a WD Lipschitz metric g. If the metric g happens to be C 1 then C is conjugate to a rotation 2 in Diffc.S /.

Sketch of proof First consider the case when R is a finite-order rotation. In this case, the diffeomorphism C is an element of finite order, say of order n. If we take any 2 Pn 1 i Riemannian metric g in S and we consider the average metric g .C / g , 0 D i 0 0 it is easy to check that g is a Riemannian metric that is invariant under CD. It follows from the uniformization theorem of Riemann surfaces that g is conformally equivalent to the standard metric in S2 . This implies that C is conjugate to a conformal map of 2 finite order of S (an element of PSL2.C/) and this easily implies that C is conjugate to a finite order rotation in SO.3/. Now consider the case when R has infinite order (an irrational rotation). We want to imitate the proof of the finite order case: let g0 be an arbitrary Riemannian metric 1 Pn i and consider the sequence of average metrics gn n i 0.C /g0 . If an appropriate D D subsequence of gn converges to a metric g, the metric g would be invariant under C f g and we would be able to conclude that C is conjugate to a rotation arguing in the same way as in the finite order case. The problem is that to obtain such a convergent subsequence, we need the sequence .C n/ .g / to not behave badly. f 0 g The idea then is to obtain bounds in the derivatives of powers of C which would assure that the sequence of average metrics gn have a nice convergent subsequence. Therefore, we are going to show that D.C n/ K, for some fixed constant K and k k Ä every integer n.

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Suppose such a constant K does not exist. Then there is a subsequence C mn such that f g D.C mn / . Passing through a subsequence, we can assume that the sequence k k ! 1 of rotations Rmn is convergent; let us assume it converges to the identity (if Rmn converges to another rotation, a similar argument works). Furthermore, we can assume kn mn e mn that D.C / e and that dC .R ; Id/ n , for the constants n and kn in k k 1 Ä Theorem 2.3.

mn Using Theorem 2.3, we can ﬁnd a ﬁnite set S s1; s2;:::; sl such that lS .R / kn mn QDln f g Ä for every n and so we can express R j 1 sij for some integers ln kn . If we D D Ä take a constant L such that for each generator D.ˆ.sm// L for every m l , we k k Ä Ä obtain

Â ln Ã ln mn Y Y kn g.n/ DC D ˆ.si / D.ˆ.si // L : D k k D j Ä k j k Ä j 1 j 1 D D This gives us a contradiction: on the one hand the sequence D.C mn / diverges kn k k superexponentially (greater than ee ), and on the other hand, it diverges at most exponentially (less than Lkn ).

It is not difﬁcult to show this implies gn has a convergent subsequence converging to a metric that a priori might be not smooth; see the proof of Lemma 2.1 for more details.

To promote the metric g in the previous proposition to an actual C 1 metric we need to have control over higher-derivatives of diffeomorphisms. We would also need an analog for higher-order derivatives of the fact that for any f; g Diff.S2/, the inequality 2 D.f g/ D.f / D.g/ holds. The purpose of the next two subsections is to k ı k Ä k kk k set up the right framework to achieve these tasks.

2.3 Norms of derivatives

In the next two sections we will define “norms” r of any element in Diffc.M / k k whose purpose is to measure how big the r –derivatives of a diffeomorphism are. We are going to use the word “norm” to refer to them but the functions r are not in k k any sense related to the various definitions of norms in the literature. This and the next subsection are technical and might be a good idea to skip them in a first read. If the reader decides to do that, he/she should have in mind that the r –norm r of k k a diffeomorphism is an analog of D. / for the r –derivative and to take a look at k k Lemma 2.11, where using the chain rule, we will obtain for any f; g Diffc.M / a 2 bound for f g r in terms of the r –norms of the diffeomorphisms f and g. k ı k

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The group Diff.M / has a natural topology known as the “weak topology”. This topology is defined as follows: take a locally finite covering by coordinate charts .Ui; i/ i of M . f g For every diffeomorphism f Diffc.M /, every compact set K Ui such that 2 f .K/ Uj , every integer r 0 and every real number > 0, there is a neighborhood i;j N .f / of f defined as the set of all diffeomorphisms g Diff.M / such that K; 2 g.K/ Uj and such that for every 0 k r , Ä Ä k 1 k 1 D .j f / D .j g / ; k ı ı i ı ı i k Ä where for any function f Rn Rn , Dk .f / denotes any k –partial derivative of a W ! component of f .

The topology induced in Diffc.M / as a subset of Diff.M / happens to be metrizable; Diffc.M / is actually a Frechét manifold (see Hirsch[14] for more details about this topology). We will denote by “d ” any metric which induces such topology in C 1 Diffc.M /.

For the proof of Lemma 2.1 we will need our norm r to have two properties. The k k first property is that the r –norm should be able to tell when the r –derivatives of a sequence of diffeomorphisms fn is diverging or not. The second property is to have f g a bound for the r –norm of f g in terms of the r –norms of f and g. Based on these ı needs and the definition of the topology of Diffc.M /, we will define for an integer r 1 the r –norms in Diffc.M / as follows. Let us start first with the case when M Rn is an open subset. In this case, we define Â for any compactly supported diffeomorphism f Diffc.M / and for any r 1, 2 ˇ r ˇ ˇ @ f ˇ f r maxˇ ˇ: k k D @xi1 @xi2 ::@xir The maximum is taken over all the partial derivatives of the different components of f and over all the points x M . 2 In order to define r for an arbitrary manifold, we use a covering set Ui i of M k k f g by coordinate charts satisfying the following properties.

n (1) Each Ui is diffeomorphic to a closed ball of R by some diffeomorphism i .

(2) .int.Ui/; i/ is a system of coordinate charts for M .

(3) Every compact set K intersects a ﬁnite number of Ui .

The existence of such covering follows easily from the paracompactness of M .

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Let us ﬁx a compact set K in M . After choosing an arbitrary covering with the properties above, we deﬁne for every f DiffK .M / 2 1 f r;K sup j f i r : k k D i;j k ı ı k

The supremum is taken over all .i; j / such that Ui K ∅, Uj K ∅, and such 1 \ ¤ \ ¤ that the expression j f is well defined. The set of .i; j / satisfying these ı ı i properties is finite and therefore the above expression is always finite.

Remark 2.7 Whenever the compact set K is clear in our context we will suppress the subindex K and denote r;K by r , but it is important to take into account k k k k that r is depending on K. k k

Lemma 2.8 Let fn be a sequence in DiffK .M /. Suppose there exist constants C and Cr (for every r 1) such that

(1) D.fn/ C , k k Ä (2) fn r Cr for n sufﬁciently large (depending on r ). k k Ä

Then there is a subsequence of fn converging in the C topology to a diffeomor- f g 1 phism f satisfying D.f / C and f r Cr . k k Ä k k Ä

Proof From the bound D.fn/ C , we conclude that the sequence fn is equicontin- k k Ä uous, and by the Arzela–Ascoli theorem we obtain a subsequence fn0 of fn converging uniformly to a continuous map f K M . W ! Similarly, for a fixed integer r 1, the inequality f r Cr implies there is a k n0k Ä subsequence f of f such that the first .r 1/ derivatives of f converge uniformly n00 n0 n00 in any fixed coordinate chart intersecting K. Using Cantor’s diagonal argument, we obtain a subsequence fn000 of fn00 such that all the r –derivatives of fn000 converge uniformly, for every coordinate chart involving K. This implies that the map f is C 1 .

Using that D.fn/ C , we can obtain that f is bi-Lipschitz and therefore invertible. k k Ä In conclusion f is a diffeomorphism and satisﬁes D.f / C . k k Ä

Observe that even though our deﬁnition of the r norms depends strongly on the k k choice of coordinates charts, the previous statement is independent of the particular choice.

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2.3.1 Metrics In the proof of Lemma 2.1, we will also need to be able to measure when a sequence of metrics gn is “diverging” in the C 1 sense. P n For a metric g gi;j dxidxj deﬁned in an closed sets U R , we deﬁne D i;j Â 8 max gi;j .x/ for r 0;

In the same spirit of Lemma 2.8, if a sequence of metrics gn defined in a closed set n U R satisfies that gn r Dr for some fixed constants Dr , it is easy to show k k Ä that gn has a subsequence converging to a C 1 metric g uniformly in each compact set. We will need to make use of the following fact, which is an easy consequence of the chain rule:

Lemma 2.9 Let U , V be closed sets of Rn . Let g be any Riemannian metric deﬁned in U and let H V U be a diffeomorphism. Suppose for every r 1, there exist W ! constants Cr , Dr such that

(1) g r Dr , k k Ä (2) H r Cr . k k Ä Then there are constants D , just depending on C and D for k r 1, such that r0 k k Ä C H .g/ r D for every r . k k Ä r0

Remark 2.10 H .g/ denotes the pullback of g under H .

2.3.2 Main inequality The next lemma is the analog for higher derivatives of the inequality D.f g/ D.f / D.g/ that we announced previously. k ı k Ä k kk k

Lemma 2.11 For every integer r 2, there exist constants Kr such that for any f; g Diffc.M / the following inequality holds: 2 r 1 f g r Kr . max f l g l / C : k ı k Ä 1 l rfk k C k k g Ä Ä Proof We will ﬁrst consider the case when M Rn . Applying the chain rule r times D to each of the components of the partial derivatives of f g we obtain a sum of terms. ı Each term in this sum is the product of at most r 1 partial derivatives of some fi C

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and gj (fi , gj denote components of f and g). The number of terms in this sum is at most a constant Kr (just depending on r and the dimension of M ) and each of the terms is less than f + g for some 1 l r , therefore k kl k kl Ä Ä Ár 1 C f g r Kr max f l g l : k ı k Ä 1 l rfk k C k k g Ä Ä For an arbitrary manifold M , we take a covering by coordinate charts .Ui; i/ as 1 in the deﬁnition of the r –norm. For every pair .i; j / such that ..f g/ .Uj // 1 ı \ Ui ∅ and every coordinate chart .Uk ; k / such that f .Uk / intersects the set ¤ 1 ..f g/ .Uj // Ui , we have that ı \ 1 1 1 j f g r j f g r k ı ı ı i k D k ı ı k ı k ı ı i k Ár 1 1 1 C Kr max j f k l k g i l Ä 1 l rfk ı ı k C k ı ı k g Ä Ä Ár 1 C Kr max f l g l : Ä 1 l rfk k C k k g Ä Ä In the previous equation, the ﬁrst inequality follows from the Rn case proved previously. We can conclude the following: Ár 1 1 C f g r sup j f g i r Kr max f l g l : k ı k D i;j k ı ı ı k Ä 1 l rfk k C k k g Ä Ä 2.4 Proof of Lemma 2.1

The following lemma is the only place where we use Militon’s theorem. Roughly, the lemma claims that there exists a compact set K Diffc.N / such that for any sequence hn converging to the identity and n sufﬁciently large, ˆ.hn/ is as close to K as one wants. More precisely:

Lemma 2.12 Let M and N be two manifolds, let K be any compact subset of M , and let ˆ Diffc.M / Diffc.N / be a group homomorphism. There exists a compact W ! set K N and constants C and Cr such that for every sequence hn in DiffK .M / 0 Â such that lim dC .hn; Id/ 0; n 1 D !1 the following hold for n sufﬁciently large:

(1) ˆ.hn/ is supported in K0 .

(2) D.ˆ.hn// C . k k Ä (3) ˆ.hn/ r Cr . k k Ä

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Remark 2.13 How large n has to be in order for ˆ.hn/ r Cr to hold might k k Ä depend on each particular r . The r –norms r in N are deﬁned with respect to the k k compact set K0 in (1).

In the proof of Lemma 2.12 we will use the following technical fact, which is a consequence of Lemma 2.11.

Lemma 2.14 Fix an integer r 2. Let Ki be the constants deﬁned in Lemma 2.11 and s1; s2;:::; sk be elements of Diffc.M /. Let L be a constant such that

(1) sj i L for every 1 i r and 1 j k , k k Ä Ä Ä Ä Ä (2) max Ki; 2 L. 1 i rf g Ä Ä Ä Then the following inequality holds:

.r 2/2k s1 s2 s r L : k ı ı k k Ä C Proof The proof goes by induction on k ; the case k 1 is obvious. Suppose it holds D for an integer k . Using Lemma 2.11, we have that Ár 1 C s1 s2 sk 1 r L max s1 s2 sk i sk 1 i k ı ı C k Ä 1 i rfk ı k C k C k g Ä Ä Ár 1 C L max s1 s2 sk i L Ä 1 i rfk ı k C g Ä Ä 2k 2k L.L.r 2/ L/r 1 L.2L.r 2/ /r 1 Ä C C C Ä C C 2k 1 2k 1 L.Lr 1L.r 2/ C / L.r 2/ C .r 2/: Ä C C D C C C Using that the inequality .r 2/2k 1 .r 2/ .r 2/2.k 1/ holds for r 1, the C C C C Ä C C results follows for k 1. C

Proof of Lemma 2.12 Existence of K 0 The proof goes by contradiction. Suppose such a compact set K N does not exist. Using Cantor’s diagonal argument one 0 can ﬁnd a sequence hn Id such that the supports of ˆ.hn/ are not contained in ! any compact set. Passing to a subsequence we can assume that dC .hn; Id/ n 1 Ä for the sequence n in Theorem 2.3. Using Theorem 2.3 we obtain a ﬁnite set f g S s1; s2;:::; s such that the sequence hn is contained in the group gen- D f l g f g erated by S . Applying ˆ the same is true for the sequence ˆ.hn/ and S f g 0 D ˆ.s1/; ˆ.s2/; : : : ; ˆ.s / , therefore the support of all the ˆ.hn/ lies in the compact f l g set K00 given by the union of the supports of the elements in S 0 , which gives us a contradiction.

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Bound for D. / Suppose that no such constant C exists. Using a diagonal k k argument we can ﬁnd a sequence hn DiffK .M / converging to the identity and 2 such that the sequence g.n/ D.ˆ.hn// diverges. Furthermore, passing to a WD k k subsequence we can assume dC .hn; Id/ n and limn log.g.n//=kn , for 1 Ä !1 D 1 the constants n and kn deﬁned in Theorem 2.3.

Using Theorem 2.3 we can ﬁnd a ﬁnite set S s1; s2;:::; s in Diffc.M / generating D f l g a group containing all the hn and such that lS .hn/ kn for every n. Therefore, we Qln Ä have hn j 1sij for some constants ln kn . If we take a constant L such that D D Ä D.ˆ.si// L for every generator si we get that k k Ä ln ln Y Y ln kn g.n/ D.ˆ.hn// D.ˆ.si // D.ˆ.si // L L : D k k D j Ä k j k Ä Ä j 1 j 1 D D

Therefore the inequality log.g.n//=kn log.L/ holds, contradicting the fact that Ä log.g.n//=kn diverges.

Bound for r The proof is similar to the previous case, but we will need to use k k Lemma 2.14. Suppose that for a ﬁxed integer r such constant Cr does not exist. In that case, we can ﬁnd a sequence hn converging to the identity and such that g.n/ ˆ.hn/ r diverges. Passing to a subsequence, we can further assume that WD k k dC .hn; Id/ n and also that 1 Ä log log.g.n// (1) lim : n kn D 1 !1 For the sequences .n/ and .kn/ in Theorem 2.3.

Using Theorem 2.3, we can find a finite set S generating a subgroup of Diffc.M / where the inequality lS .hn/ kn holds for every n. Let L be a constant such that Ä ˆ.g/ r L for every g S and which satisfies the hypothesis of Lemma 2.14. k k Ä 2 Using Lemma 2.14, we have that

.r 2/2kn g.n/ ˆ.hn/ r L : D k k Ä C Taking logarithms in both sides we obtain

log log.g.n// 2kn log.r 2/ log.log.L//; Ä C C which contradicts (1).

Finally, we will formulate Lemma 2.1 one more time and ﬁnish its proof.

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Lemma 2.1 Let M and N be C manifolds and let ˆ Diffc.M / Diffc.N / be 1 W ! a group homomorphism. Suppose that hn is a sequence in Diffc.M / supported in a compact set K and such that

lim dC .hn; Id/ 0: n 1 D !1 Then ˆ.hn/ has a convergent subsequence, converging to a diffeomorphism H , f g which is an isometry for a C 1 Riemannian metric on N .

Proof From Lemma 2.12 we conclude that given a fixed integer r 1, for n large enough the inequality ˆ.hn/ r Cr holds. Using Lemma 2.8 we can extract a con- k k Ä vergent subsequence from ˆ.hn/ which is converging to a C diffeomorphism H . f g 1 Even more, taking powers of hn we have the following. For every fixed integer k , the k k sequence hn Id and then the inequality ˆ.hn / r Cr holds for n large enough. ! k k k Ä Therefore we can conclude that H r Cr for every integer k . k k Ä Fix a Riemannian metric g0 on N and consider the sequence of metrics n 1 X k gn .H / g : D n 1 0 C k 0 D k (The notation .H /g0 denotes the pullback of the metric g0 under the diffeomor- k phism H .) Next, we will show that the sequence gn has a subsequence converging to a C 1 metric g, which is invariant under H . To simplify the rest of the proof we will use the following notation. Take coordinate charts .Ui; i/ of N as in the definition of the r norms (see Section 2.3). For any i k k n metric g in N , let g be the metric defined in the closed set i.Ui/ R obtained by pullback, more concretely gi . 1/ g. WD i Going back to our proof, observe that a diagonal argument shows that gn has a convergent subsequence if and only if in any coordinate chart, the corresponding sequence of metrics has a convergent subsequence. More concretely, we just need to i show that for any fixed integer i , the sequence of metrics g n (as defined in the f ng previous paragraph) has a convergent subsequence. i 1 Pn k i Observe that gn n 1 k 0..H /g0/ . From (1) in Lemma 2.12 the diffeomor- D C D S k phism H is supported in a compact set, therefore for each fixed i the set k H .Ui/ is contained in the union of a finite number of Uj and so we can assume it intersects l nontrivially the sets Ul for 1 l j . We define Dr max1 l j g r . Ä Ä WD Ä Ä k k k For every integer k and each x Ui , the point H .x/ lies in the interior of U for 2 l some 1 l j . Thus we have that ..H k / g/i . H k 1/ gl in a small Ä Ä D l ı ı i

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k 1 l ball around x. Using the inequalities l H i Cr and g Dr , we can k ı ı k Ä k ki k Ä apply Lemma 2.9 to obtain constants Dr0 such that 0 ..H /g/ r Dr0 for every Äi k k Ä integer r 1. The previous inequality implies that gn r Dr0 for every n 1 and i k k Ä therefore our sequence g n has a convergent subsequence as we wanted. f ng In conclusion, we can extract from the sequence gn a subsequence converging to a C 1 metric g. To ﬁnish our proof we just need to show that the metric g is nondegenerate and invariant under H . Nondegeneracy As the inequality D.H k / C holds for every integer k , we k k Ä conclude that for each vector v TM , each of the metrics gn satisfy the inequality 2 g0.v; v/ 2 gn.v; v/ C g .v; v/: C 2 Ä Ä 0

This implies that any convergent subsequence of gn converge to a nondegenerate f g metric and so our g is nondegenerate. Invariance under H For any vector v TN , we have 2 n 1 .H C /g.v; v/ g.v; v/ g.v; v/ H gn.v; v/ gn.v; v/ k k .C 1/ k k : k k D n 1 Ä C n 1 C C Taking limits for our subsequence in the previous inequality we obtain that the metric g is invariant under H .

As a corollary, we obtain a proof of Theorem 1.2 in the case N is noncompact.

Corollary 2.15 Let M and N be C 1 manifolds and suppose that N is noncompact. Then any group homomorphism ˆ Diffc.M / Diffc.N / is weakly continuous. W !

Proof Let K be any compact set in M . Arguing by contradiction, suppose hn is a sequence converging to the identity in DiffK .M / but such that ˆ.hn/ is not converging to the identity. We can suppose that dC .ˆ.hn/; Id/ C > 1 for every n 1. By 1 Lemma 2.1 there is a subsequence of ˆ.hn/ converging to an isometry H in N and by (1) in Lemma 2.12 the isometry H should be the identity outside a compact set K0 of N , therefore H is the identity everywhere contradicting that dC .ˆ.hn/; Id/ C > 1. 1 3 Continuous case

3.1 Notation

Let M and N denote two C 1 manifolds of dimension m and n respectively. For the rest of this section we will assume that N is closed. If U is an open subset of M , we

Geometry & Topology, Volume 19 (2015) Continuity of homomorphisms of diffeomorphism groups 2133 deﬁne GU f Diffc.M / supp.f / U : D f 2 j g We recall the deﬁnition of “weak continuity” given in Section 1.

Deﬁnition 3.1 For any compact subset K M , let DiffK .M / denote the group of dif- Â feomorphisms in Diffc.M / supported in K. A group homomorphism ˆ Diffc.M / W ! Diffc.N / is weakly continuous if for every compact set K M , the restriction Â ˆ of ˆ to DiffK .M / is continuous. jDiffK .M / Remark 3.2 If M is closed, this deﬁnition is equivalent to ˆ being continuous in the usual sense.

3.2 Weak continuity vs continuity

The concept of weak continuity for a group homomorphism ˆ Diffc.M / Diffc.N / W ! is not equivalent to our homomorphism ˆ being continuous in the weak topology of Diffc.M / and one should not expect such continuity to hold. For example, take the open unit ball B Rn . The homomorphism induced by the inclusion i B Rn is a W ! group homomorphism which is not a continuous map in the weak topology.

Take a sequence of diffeomorphisms fn supported in small disjoint balls Bn contained in B such that each Bn is centered at a point pn and has radius rn in such a way that the sequence pn converges to a point in @B , the radii rn 0 and such that x ! Dp .fn/ 2 Id. The sequence fn converges to the identity in Diffc.B/ in the weak n D topology because restricted to every compact set K B , the sequence fn is the identity n for large n. Nonetheless fn does not converge in Diffc.R / and therefore ˆ is not continuous. In this section, we will assume that ˆ is always weakly continuous. The main purpose of this section is to establish the following result:

Theorem 3.3 If ˆ Diffc.M / Diffc.N / is a nontrivial weakly continuous homo- W ! morphism of groups, then dim.M / dim.N /. If dim.M / dim.N /, then ˆ is Ä D extended topologically diagonal.

The deﬁnition of “extended topologically diagonal” will be given in Deﬁnition 3.12. The assumption of weak continuity is very strong for a homomorphism between groups of diffeomorphisms. As a proof of that, we recall the following classic result (see Montgomery and Zippin[19, Chapter 5]):

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Theorem 3.4 (Montgomery, Zippin) Let ˆ be a continuous homomorphism from a ﬁnite-dimensional Lie group G to the group Diffc.M / of C 1 diffeomorphisms of a manifold M . Then the map M G M given by .x; g/ ˆ.g/x is a C W ! D 1 map in both variables .x; g/ simultaneously.

Corollary 3.5 If ˆ Diffc.M / Diffc.N / is weakly continuous and X is a com- W ! pactly supported vector field in M generating a flow ft Diffc.M /, then ˆ.ft / is a 2 flow in Diffc.N / generated by some vector field Y supported in N .

Proof H.t/ ˆ.ft / defines a homomorphism from R to Diff.M /. H.t/ is contin- D uous by the weak continuity of ˆ and therefore is a C 1 flow by Theorem 3.4. We obtain that ˆ.ft / is generated by some vector field Y in N .

In our proofs, we will use the following known fact. For a proof of this fact, see[4].

Lemma 3.6 (Fragmentation property) Let M be a manifold and B be an open cover of M ; then Diffc.M / is generated by the following set:

C f Diffc.M / such that supp.f / B0 for some B0 B : D f 2 2 g In the next two subsections we will construct some examples of homomorphisms between groups of diffeomorphisms when dim.M / dim.N /. D 3.3 Examples

When dim.M / dim.N /, there are two fundamental ways of constructing examples D of homomorphisms of the type ˆ Diffc.M / Diffc.N /. The ﬁrst one is to embed W ! a ﬁnite number of copies of M into N and to act in such copies in the obvious way (topologically diagonal). The second one consists of homomorphisms coming from lifting diffeomorphisms of M to a covering N . The main theorem of this section shows that if ˆ is continuous, all possible homomorphisms are combinations of these two examples. We will discuss these two fundamental examples in more detail.

3.3.1 Topologically diagonal

Deﬁnition 3.7 A homomorphism ˆ Diffc.M / Diffc.N / is topologically diagonal W ! if there exists a ﬁnite collection of disjoint open sets Ui N and diffeomorphisms Â i M Ui , so that for every f Diffc.M / we have W ! 2 1 i f .x/ if x Ui; ˆ.f /.x/ ı ı i 2 D x otherwise.

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Observe that if M N and N is connected, then M has to be necessarily noncompact ¤ for the previous definition to make sense. The fact that Ui i is a finite collection is f g also necessary, even though we may be able to embed an infinite collection of manifolds M in a closed manifold N , if the collection is not finite, the homomorphism ˆ will not be well defined. To better illustrate this last assertion, observe the following

n Let M R and suppose there is an infinite collection of open sets Ui N giving D rise to a homomorphism ˆ as in Definition 3.7. Take a diffeomorphism f supported in the unit ball around the origin in Rn , such that f .0/ 0 and Df .0/ 2 Id. D D Using the notation of i as in Definition 3.7, consider the sequence i.0/ pi in N . D By assumption, pi i is an infinite sequence and therefore has an accumulation point p S f g in N Ui . We have that D.ˆ.f //.pi/ 2 Id, and by continuity we have that n i D D.ˆ.f //.p/ 2 Id. It is not difficult to see by the compactness of N that there is D a another set of points qi p such that ˆ.f /qi qi and that D.ˆ.f //.qi/ Id, ! D D which implies that D.ˆ.f //.p/ Id, a contradiction. D 3.3.2 Finite coverings Many times when N is a finite covering of M there is a homomorphism ˆ Diffc.M / Diffc.N / obtained by lifting a given f Diffc.M / W ! 2 to an appropriate diffeomorphism fz of N . For example, if M and N are closed hyperbolic surfaces, every diffeomorphism of M isotopic to the identity can be lifted to the universal covering H2 (hyperbolic plane). Even more, it can be lifted to a unique homeomorphism f H2 H2 such that f restricts to the identity in @H2 S1 . 0W ! 0 D This lift commutes with all the covering translations, and so one can project f 0 to a diffeomorphism fz of our covering surface N . The uniqueness of these lifts ensures that fg f g. Therefore, we obtain a homomorphism ˆ Diffc.M / Diffc.N /. D zz W ! Unfortunately, this is not true for all covering maps N M . For example, if W ! we let M N S1 and S1 2 S1 , the obvious 2–cover, any of the lifts of the D 1D W ! rotation r2 in S of order 2, has order 4, and therefore it is impossible to construct a homomorphism ˆ that commutes with p. The obstruction as we will see, comes from 1 the fact that 1.Diffc.S // Z is generated by a full rotation around the circle and Š that S1 2 S1 corresponds to the subgroup 2Z of Z. W ! In a more precise way, we have the following situation. Let G Diffc.M /, H be D the group consisting of lifts of elements of G to N and L be the group of deck transformations corresponding to the covering map N M , we have an exact W ! sequence of discrete groups (2) e L H G e: ! ! ! ! We want to understand when this sequence splits.

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For sake of completeness, we mention a useful criterion for proving this. For a proof, see Brown[5, page 104].

Lemma 3.8 The sequence e L H G e splits if and only if the two ! ! ! ! following conditions are satisﬁed.

The natural map H Out.L/ lifts to Aut.L/. ! The element in H 3.G; Z.L// deﬁned by the exact sequence is trivial (Z.L/ denotes the centralizer of L).

As a consequence of Theorem 1.2 (whose proof will be given in Section 5) our homomorphism ˆ G H is continuous (when M and N are closed) and therefore W ! showing that the previous sequence splits discretely is equivalent to show it splits continuously. Therefore, we just need to understand in which case there exists a continuous homomorphism ˆ G H which splits our sequence. W ! Observe that H is a covering of G , and so the existence of such continuous section ˆ is equivalent to the covering H being trivial: H as a topological space should consist of the union of disjoint copies of G . This happens if and only if every element of L lies in different components of H . In conclusion, our sequence splits if there is no deck transformation L isotopic to the identity in H . In conclusion, our exact sequence (2) splits if and only if there is no deck transformation L of the covering N M , which is isotopic to the identity in Diffc.N /. W ! Observe that if there is a path H connecting the identity to a nontrivial deck 2 transformation L, projecting down to G we obtain a loop 0 representing a nontrivial element in 1.G; e/, the fundamental group of G based at the identity. To obtain a more precise statement of when this happens, we will need the following deﬁnition.

Take an arbitrary point x M . There is a natural evaluation map Ex Diffc.M / M 2 W ! given by Ex.f / f .x/ and so we have a natural homomorphism of fundamental D groups Ex 1.Diffc.M /; e/ 1.M; x/: W ! Observe the following fact:

Proposition 3.9 A closed loop 1.G; e/ has a lift to H connecting the identity 2 to a nontrivial deck transformation if and only if Ex. / does not lift to a closed loop in N .

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Proof In one direction, if a loop 1.G; e/ satisﬁes Ex . / 1.N /, the lift of 2 62 Ex . / to N is not a closed loop. Therefore, the lift of to H Diffc.N / starting at the identity has to have the other endpoint in a nontrivial lift of the identity of Diffc.M /, that is, a nontrivial deck transformation. The other direction is trivial. Summarizing all the previous discussion, we have the following criterion:

Lemma 3.10 Let N M be a covering map. Then the following statements are W ! equivalent:

(1) There exists a unique continuous homomorphism Diffc.M / Diffc.N / W ! such that for every f Diffc.N /, the diagram 2

.f / N / N

M / M f commutes.

(2) The inclusion E .1.Diffc.M /; e// 1.N / holds. x Observe that if ft is a path in Diffc.M / representing a loop ˛ of 1.Diffc.M /; e/ based at the identity and s is a loop based at x representing an element of 1.M /, then the family of paths ft . s/ gives us a homotopy between the loops and 1 f g .Ex.ft // .Ex.ft //. This shows that the image of Ex is a central subgroup of 1.M /. In particular, we obtain the following useful corollary:

Corollary 3.11 If N M is a covering map and 1.M / is centerless, then there W ! is a homomorphism Diffc.M / Diffc.N / as in Lemma 3.10. W ! 3.4 Extended topologically diagonal

Based on the previous two examples we make the following deﬁnition:

Definition 3.12 A homomorphism ˆ Diffc.M / Diffc.N / is extended topologi- W ! cally diagonal (see Figure 1) if there exists a finite collection of disjoint open sets Ui N and a set of finite coverings i Mi M , together with a collection of Â W ! diffeomorphisms i Mi Ui such that for every f Diffc.M /, W ! 2 1 i .f / .x/ if x Ui; ˆ.f /.x/ ı i ı i 2 D x otherwise, where Diffc.M / Diffc.Mi/ are the homomorphisms coming from the finite iW ! coverings i as in the previous subsection.

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Remark 3.13 The coverings have to be ﬁnite in order for ˆ to be an actual homomorphism into Diffc.N /. See the comments at the end of Section 3.3.1.

We will illustrate the previous deﬁnition with the following example:

M2 M1

Figure 1: An extended topologically diagonal embedding

Example 3.1 Let M be the once-punctured 2–dimensional torus and let N be the closed orientable surface of genus 3. Take M1 to be a copy of M and M2 to be the twice-punctured torus (which is a double cover of M ) as in Figure 1. We embed M1 and M2 into N as illustrated in Figure 1 and obtain a homomorphism ˆ Diffc.M / Diffc.N /. W ! We are now in position to give a proof of the main result of this section.

3.5 Proof of Theorem 3.3

In order to prove Theorem 3.3, we will need to make the following deﬁnitions:

Definition 3.14 For any group G Diffc.N /, we define Â supp .G/ p N such that g.p/ p for some g G : 1 D f 2 ¤ 2 g For each x M , let us define 2 \ Sx supp .ˆ.GB //; D 1 r r>0

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where Br is an open ball of radius r around x. We construct the sets Sx thinking of constructing the maps i in the definition of extended topologically diagonal. We would like to show that Sx consists of a discrete set of points and that the correspondence Sx x give us a collection of coverings from some open sets of N to M . ! Let N N fix.ˆ.Diffc.M /// be the set of points in N that are not fixed by the 0 WD n action of Diffc.M /.

Lemma 3.15 Let ˆ Diffc.M / Diffc.N / be a nontrivial homomorphism. For W ! every x M we have the following. 2 (1) Sx is a nonempty set.

(2) For any f Diffc.M /, ˆ.f /Sx Sf .x/ . S 2 D (3) N 0 Sx . D x M 2 Proof We start with the proof of item (3). If p N , there is an element f0 in 2 0 Diffc.M / such that ˆ.f0/.p/ p. Using the fragmentation property, we can ex- ¤ press f as a product of diffeomorphisms supported in balls of radius one and so we obtain a diffeomorphism f1 supported in B1 M such that ˆ.f1/.p/ p. We ¤ can iterate this inductive procedure and construct a diffeomorphism fn Diffc.M / 2 supported in a ball Bn of radius at most 1=n, such that Bn Bn 1 and such that T ˆ.fn/p p. We deﬁne x Bn . It is clear that p Sx , so item (3) follows. ¤ WD n 2 Item (2) follows by conjugating f and applying ˆ. Item (1) is a direct consequence of (2) and (3).

The following two lemmas are technical lemmas needed for the proof of Lemma 3.18.

Lemma 3.16 Given x0 M and U M an open set containing x0 , there exists a 2 Â neighborhood Nx of x0 in M and a continuous map ‰ Nx GU such that for 0 W 0 ! every x Nx we have that ‰.x/.x0/ x. 2 0 D n Proof Using coordinate charts is enough to check the case when M R and x0 0. n D D For 1 i n, let Xi be vector fields in R supported in U such that Xi.0/ i is a Ä Ä i f g linearly independent set of vectors. If ft are the flows generated by Xi , we have that i f GU . t 2 n n n We define for t .t1; t2;:::; tn/ . ; / , the map . ; / R by .t/ 1 2 n D 2 W ! D ft1 ft2 ftn .0/. By our choice of Xi , for sufficiently small , is a diffeomorphism n onto a small neighborhood N0 of 0 R . 2 1 If .x/ .t1.x/; t2.x/; : : : ; tn.x//, we can define the map ‰ N0 GU by ‰.x/ D W ! D f 1 f 2 f n . It is clear that ‰ is continuous and ‰.x/.0/ x. t1.x/ t2.x/ tn.x/ D

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Lemma 3.17 Let ˆ Diffc.M / Diff.N / be a weakly continuous homomorphism W ! and let p be a point in N . There are no n 1 pairwise disjoint open subsets Ui M , C i 1; 2;:::; n 1, such that every subgroup ˆ.GU / does not ﬁx p. D C i

Proof Suppose that such a point p exists. As a consequence of the simplicity of GUi , any element of GUi can be written as a product of a finite number of flows. Therefore, i i for every 1 i n 1, there exists a flow ft supported in Ui , such that ˆ.ft / does Ä Ä C i not fix p. Observe that the vector fields Xi generating the flows ˆ.ft / satisfy that i j Xi.p/ 0. Also, as ˆ.f / and ˆ.f / commute, ŒXi; Xj 0 for every i; j . ¤ t t D Take a maximal collection of vector fields C X1; X2;:::; X in N such that D f k g X1.p/; X2.p/; : : : ; Xk .p/ are linearly independent and such that all the Xj comes from flows in different ˆ.GUi /. Observe that k must be less than or equal to n; therefore we can assume without loss of generality that if ft is any flow in GUn 1 C and Y is the vector field corresponding to the flow ˆ.ft /, then Y .p/ is a linear combination of elements in X1.p/; X2.p/; : : : ; X .p/ . f k g If we consider the open set

U q N X1.q/; X2.q/; : : : ; X .q/ are linearly independent ; D f 2 j k g the vector fields in C define a foliation F on U by the Frobenius integrability theorem. Let L be the leaf of F containing p. Observe that as Y .p/ is a linear combination of elements of X1.p/; X2.p/; : : : ; X .p/ and ŒY; Xi 0, then we have that Y is f k g D tangent everywhere to the leaves defined by C . It also follows that if q U , then 2 ˆ.gt /q U for every t . In conclusion, we have that ˆ.gt / necessarily preserves L. 2

As the diffeomorphism gt was an arbitrary flow in GUn 1 , we have obtained a non- C trivial homomorphism ‰ GUn 1 Diff.L/. Even more, the image of this map is W C ! abelian, since if Y and Z are vector fields coming from ˆ.GUn 1 /, both Y and Z commute with all the elements of C , and therefore Y and Z can beC shown to be linear combinations of vector fields in C . This contradicts the simplicity of GUn 1 . C

Lemma 3.18 If x; y are 2 distinct points on M , then Sx Sy ∅. \ D

Proof From Lemma 3.17, for any n 1 different points x1; x2;:::; xn 1 in M we C C have that Sx1 Sx2 Sxn 1 ∅. Let k be the largest integer such that there \ \ \ C D exist x1; x2;:::; xk distinct points on M such that Sx1 Sx2 Sxk ∅. We \ \ \ ¤ are going to show that k 1. D Let p be an arbitrary point in Sx Sx Sx . By Lemma 3.16, we can ﬁnd 1 \ 2 \ \ k pairwise disjoint neighborhoods Nx of M and continuous maps ‰i Nx Diffc.M /, i W i !

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such that ‰i.y/.xi/ y for every y Nx . Even more, we can suppose that the D 2 i support of the elements in the image of ‰i and ‰j have disjoint support if i j . ¤ Q k Let V Nx M . We deﬁne the map ‰ V N given by D i i Â W ! ÂY Ã ‰.y1; y2;:::; y / ˆ ‰i.yi/ .p/: k D i

We have that ‰ is continuous because ˆ and all the ‰i are continuous. Observe Q Q that . ‰i.yi//.x / y for every 1 l k , therefore ˆ. ‰i.yi//.p/ Sy i l D l Ä Ä i 2 1 \ Sy Sy by property .2/ in Lemma 3.15. By maximality of k , we obtain 2 \ \ k that ‰ is injective. It follows then that dim.M k / dim.N / and therefore k 1 and Ä D dim.M / dim.N /. D

As a corollary of the last paragraph we obtain:

Corollary 3.19 If ˆ Diffc.M / Diffc.N / is a weakly continuous homomorphism W ! and dim.N / < dim.M /, then ˆ is trivial.

Corollary 3.20 If ˆ Diffc.M / Diffc.N / is a weakly continuous homomorphism W ! of groups and dim.M / dim.N /, then for each p N we have that ˆ.Diffc.M //p D 2 0 is an open set of M .

Using the previous corollary, we obtain that for our set N 0 N fix.ˆ.Diffc.M /// S WD n defined before Lemma 3.15, we have that N Ui where each Ui is an open set 0 D i defined as Ui ˆ.Diff.M //pi for some point pi N . Observe that each Ui is a WD 2 0 connected open set where the action of Diffc.M / is transitive. To finish the proof of Theorem 3.3, we just need to show the following:

Lemma 3.21 If ˆ.Diffc.M // acts transitively on N , then there is a C 1 covering map N M such that for every f Diffc.M / the following diagram commutes: W ! 2 ˆ.f / N / N

M / M f

In other words, ˆ as in Section 3.3.2. D

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Proof Let N M be the map deﬁned by .p/ x if p Sx . Observe that the W ! D 2 map is well deﬁned by Lemma 3.18 and is surjective by our transitivity assumption. Next, we will show that is locally a C 1 diffeomorphism.

Using Lemma 3.16, we can find a neighborhood Nx of M and a continuous map ‰ Nx Diffc.M / such that ‰.y/.x/ y for every y Nx . We can define the map W ! D 2 Nx N given by W ! .y/ ˆ.‰.y//.p/: D Observe that is continuous because ˆ is weakly continuous and injective because .y/ Sy and Lemma 3.18. We also have that Id which implies that is 2 ı D a covering map. To show that is a C 1 map it is enough to show that is a C 1 diffeomorphism onto its image. The fact that is smooth follows from Theorem 3.4: it is enough to prove that the image of a C 1 embedded curve passing through x is a C 1 curve. Given , we can construct a C 1 compactly supported vector field X such that is a flow line of this vector field. The vector field X defines a flow ft and by Corollary 3.5 we have that ˆ.ft / is a C flow. Observe that .ft .x// S ˆ.ft /Sx . Using that 1 2 ft .x/ D is locally a homeomorphism for t sufficiently small we have that .ft .x// ˆ.ft /p D and therefore . / is smooth.

4 Blowups and gluing of actions

In the proof of Theorem 1.2 we are going to make use of the following two constructions on group actions on manifolds.

4.1 Blowups

Given a closed manifold N and a closed embedded submanifold L N of positive codimension, let Diffc.N; L/ be the group of diffeomorphisms of N preserving L setwise. Suppose we have a group G Diffc.N; L/. We will show how to construct Â a natural action of G on N , where N is the manifold obtained by blowing up N along L. Intuitively, the blowup N is obtained by compactifying N L by the unit normal n bundle of L in N . More concretely, ﬁx a Riemannian metric on N and let N.L/ be the unit normal bundle of our submanifold L in that metric. The normal neighborhood theorem states that the map

p0 N.L/ .0; / N W !

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defined by sending .v; t/ to the point v.t/ in N at distance t along the geodesic with initial vector v, is a diffeomorphism onto a neighborhood of L for sufficiently small . We now define formally N , the blowup of N along L, as

N .N.L/ Œ0; // .N L/=p0: D [ n The blowup N is a compact manifold with N.L/ as its boundary and there is a canonical smooth map N N . The smooth structure on the blowup N does W ! not depend on the metric, N can be also described as the compactiﬁcation of N L S n by the set of directions up to positive scale in q L.TqN=TqL/. 2 In order to describe the smooth structure on N in more detail, suppose for the rest of this section that L consists of a single point p (the more general case can be treated in a similar way).

Let x1; x2;:::; xn be coordinate charts around p in N , where xi.p/ 0 for every D 1 i n and let v be a direction up to positive scale in Tp.N /. Without loss of Ä Ä generality, assume that in our coordinate charts we have that v v1; v2; : : : ; vn 1; 1 . D h i Then, there is a corresponding set of coordinate charts 1; 2; : : : ; n 1; xn around v in N , where v corresponds to the point i vi , xn 0 and the change of coordinates D D for xn > 0 is given by xj xnj . D The next lemma is the main fact that we need about blowups of actions and implies that for a given group action ˆ G Diffc.N /, preserving a submanifold L, there is W ! a corresponding action ˆ G Diffc.N / in the corresponding blowup N , more 0W ! precisely:

Lemma 4.1 There is a “blowup” map Diffc.N; L/ Diffc.N / such that for W ! every h Diffc.N; L/ the following diagram commutes: 2 .h/ N / N

N / N h Even more, is a group homomorphism.

Proof For any h in Diffc.N / we will define .h/ as follows: if x N @N we 2 n define .h/.x/ h.x/. On the boundary @N , we can define .h/ as the projectiviza- D tion (up to positive scale) of the map Dh.p/ TpN TpN . It is easy to see .h/ is W ! continuous.

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We need to check that .h/ is a smooth map: suppose in our coordinate charts x1; x2;:::; xn around p in N that the diffeomorphism h is given locally by h n D .h1;:::; hn/, where p correspond to 0 R and h.0/ 0. Given a direction v 2 D (up to positive scale) at the origin, such as v v1; v2; : : : ; vn 1; 1 , there exists D h i some j such that hj .p/.vi; 1/ 0, so we can assume that hj .p/.vi; 1/ > 0. In r ¤ r our coordinate charts 1; 2; : : : ; n 1; xn explained previously, we have that .h/ is given by .h/ .h ; h ;:::; h / where D 1 2 n ( hi .xn;xn/ if i j ; hj .xn;xn/ hi .; xn/ ¤ D hj .xn; xn/ if i j ; D where 1; 2; : : : ; n 1 and xn 0. WD h i ¤ When xn 0 (in the boundary), we have that D hi.xn; xn/ hi.0/ ; 1 hi .; 0/ lim r h i : D xn 0 hj .xn; xn/ D hj .0/ ; 1 ! r h i @ Additionally, observe that we have hj .xn; xn/ hj .xn; xn/.; 1/ and that @xn D r hj .0/.; 1/ 0. r ¤ All the previous formulas imply that the derivatives of any order of .h/ exist for points in the boundary @N and therefore .h/ belongs to Diffc.N /.

We still need to show that Diffc.N; L/ Diff.N / is a group homomorphism, W ! more precisely, we need to show that for any f; g Diffc.N; L/, the equality .fg/ 2 D .f /.g/ holds. This equality is obvious for points in N @N and it follows from n the chain rule for points in @N .

For more details and other aspects about the previous blowup construction, see Kron- heimer and Mrowka[15, Section 2.5]. A related blowup for diffeomorphisms is studied in great detail in Arone and Kankaanrinta[2].

4.2 Gluing actions along a boundary

Given N1 and N2 two compact manifolds with nonempty boundary, suppose there is a diffeomorphism ˛ @N 1 @N 2 . Denote by N N 1 N 2 the manifold obtained by 1 W2 ! D [ gluing N and N along the boundary using ˛. Suppose we have actions ˆi G i W ! Diffc.N / in such a way that the actions of every element g G coincide in the 2 common boundary. More precisely, suppose that for each g G , the equality ˆ2.g/ 1 2 2 D ˛ ˆ1.g/ ˛ holds for points in @N . ı ı

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In general, if we glue ˆ1.g/ and ˆ2.g/ along the common boundary, we will not obtain a C 1 diffeomorphism of N . Nonetheless, the following theorem of Parkhe[20] gives a way to modify the actions in such a way that the resulting glued action of G on our manifold N is a truly C 1 action.

1 2 Theorem 4.2 There are homeomorphisms ‰1 and ‰2 of N and N with the following properties. i 1 i (1) For any f Diffc.N /, the map ‰if ‰ Diffc.N /. 2 i 2 1 2 (2) If f1 Diffc.N / and f2 Diffc.N / coincide in the common boundary, 2 2 1 i then the diffeomorphism f deﬁned by ‰ifi‰ in N for i 1; 2 is a C i D 1 diffeomorphism of N .

1 Thus, if we deﬁne new homomorphisms ˆ G Diffc.Ni/ by ˆ .g/ ‰iˆi.g/‰ , i0 W ! i0 D i we can glue both ˆ and ˆ to obtain a homomorphism ˆ G Diffc.N /. 10 20 0W ! 5 Proof of Theorem 1.2

In this section, we will use the results of Section 2 together with the facts discussed in Section 4 to give a proof of Theorem 1.2. This result along with Theorem 3.3 completes also the proof of Theorem 1.3. n From now on, we denote G Diffc.R /. As in Section 3, if B is an embedded open D ball contained in Rn , we denote

GB f Diffc.M / supp.f / B : D f 2 j g In the proof of Theorem 1.2 we are going to make use of the following basic fact:

Lemma 5.1 We say ˆ is weakly continuous if for some ball B M whose closure is a closed ball embedded in M . We have that the restriction ˆ G of ˆ to the j B subgroup GB is continuous.

Proof Conjugating with elements of Diffc.M / we get that for any embedded ball B0 , the homomorphism ˆ Diff .B / is continuous. From the proof of the fragmentation j c 0 lemma[4, Lemma 2.1.8], one obtains that if the sequence fn tending to Id is supported in a compact set K and C is a ﬁnite covering of K by sufﬁciently small balls, then for large n, the diffeomorphisms fn can be written as products of diffeomorphisms supported in open sets of C . Furthermore, we can achieve this with at most k factors for each open set of C (for some constant k ) and it can be done in such a way that these diffeomorphisms are also converging to the identity.

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We will proceed to give an outline of our main result, Theorem 1.2.

Outline of the proof of Theorem 1.2 The proof goes by induction in the dimension of N . Consider any open ball B whose closure is a closed ball embedded in M . By Lemma 5.1, to show that ˆ is weakly continuous, it is enough to show that ˆ is weakly continuous when we restrict to the subgroup GB . We are going to use many times the following fact: ˆ is weakly continuous if there is no sequence fn in GB , such that fn Id and ˆ.fn/ A, where A is a nontrivial ! ! isometry in some Riemannian metric on N . This fact is an immediate consequence of Lemma 2.1.

Assume such a nontrivial isometry A exists. If we take a ball B0 disjoint from B , the actions of G and A commute. If A has a nontrivial ﬁxed point set L, then G B0 B0 preserves L setwise, a manifold of lower dimension than N . Therefore, we can blowup the action at L and show by induction that ˆ is continuous.

If A acts freely instead, we will obtain an action of GB on the space N N=H , 0 0 WD where H A is the closure of the subgroup generated by A. If the group H is D h i infinite, N 0 is a manifold of lower dimension than N and we can proceed to prove the continuity by induction. If H is finite, we will show a way to replace H by a similar infinite group of isometries H to proceed as in the case where H is infinite. y In the proof of Theorem 1.2, we are going to make use of the following known fact:

Lemma 5.2 If N is a closed Riemannian manifold and H is a nontrivial closed connected subgroup of the group of isometries of N , then ﬁx.H / p N D f 2 j h.p/ p for every h H is a closed submanifold of positive codimension. D 2 g We will also need the following technical lemma:

Lemma 5.3 If ˆ Diffc.B0/ Diffc.N / is a weakly continuous homomorphism, W ! then there exists an embedded open ball B2 B1 and a point p N such that for any 2 diffeomorphism f supported in B2 , the equality ˆ.f /p p holds. D Proof Deﬁne n dim.N / 1 and consider an arbitrary point p N . For 1 i n, D C 2 Ä Ä let Ui be disjoint embedded balls contained in B1 . Using Lemma 3.17, we conclude that for at least one of these balls, which we denote by B2 , the restricted homomorphism ˆ G ﬁxes p. j B2 By Lemma 5.1, Theorem 1.2 is a direct consequence of the following:

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m Lemma 5.4 Let N be a closed manifold. If ˆ Diffc.R / Diffc.N / is a group W ! homomorphism, then ˆ is weakly continuous.

Proof We proceed by induction on the dimension of N . The proof for the case n 0 D is trivial and the case n 1 is proved in[16]. D Let B be the unit ball in Rn . Suppose that ˆ is not weakly continuous. By Lemma 2.1, there is a ball B0 B strictly contained in B and a sequence fn GB such that 2 0 fn Id and ˆ.fn/ A, where A is a nontrivial isometry for some Riemmanian ! ! metric on N . Deﬁning H A , H is a closed subgroup of the Lie group of WD h i isometries of N .

For the rest of the proof, we let B1 B be an embedded ball disjoint from B0 . Observe Â that as B0 and B1 are disjoint, the groups ˆ.GB0 / and ˆ.GB1 / commute. First we show that we can assume H acts freely:

Lemma 5.5 If H does not act freely, then ˆ is weakly continuous.

Proof Suppose that H does not act freely. We can replace our group H with a nontrivial stabilizer of some point p N and then assume that L ﬁx.H / ∅. By 2 D ¤ Lemma 5.2, L is a closed submanifold of positive codimension.

Observe that the action of ˆ.GB1 / on N commutes with the action of H and therefore L is invariant under ˆ.GB1 /. Using Lemma 4.1 we can blowup the action ˆ at L to obtain an action ˆ GB1 N . This new action preserves the boundary @N W ! m D N.S/, and therefore we obtain a homomorphism ˆ Diffc.R / Diff.@N / j@N W ! that by the inductive hypothesis is weakly continuous. To prove that ˆ is weakly continuous, we will need to deﬁne the following objects: Let N 1 and N 2 be two copies of N . Additionally, let N N 1 N 2 be the manifold 0 WD [ obtained by gluing N 1; N 2 in the obvious way. Finally, consider the homomorphisms i ˆi GB Diffc.N / equal to ˆ for i 1; 2. W 1 ! D

Observe that the previous homomorphisms deﬁne an action of GB1 on N 0 , but this action might not be smooth in the submanifold corresponding to the glued boundaries. Using the gluing theorem described in Section 4.2, we can conjugate the actions in i each N by homeomorphisms and obtain a smooth action ˆ GB Diffc.N /. 0W 1 ! 0 We are now going to make use of the following trick: we will show that ˆ0 is weakly continuous using the induction hypothesis and then we will show that the continuity of ˆ0 implies the continuity of ˆ.

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Let us show that ˆ GB Diffc.N / is weakly continuous. By Lemma 2.1, it is 0W 1 ! 0 enough to show the following: if fn GB is a sequence such that fn Id and 2 1 ! ˆ .fn/ A, where A is an isometry for N , then A is trivial. 0 ! 0 By the inductive hypothesis, the restriction of ˆ0 to the submanifold corresponding to the glued boundaries is continuous, therefore A is trivial in a codimension-one submanifold. As any orientation-preserving isometry that is trivial in a codimension- one submanifold is trivial everywhere, we obtain that A must be trivial.

Now we show that the weak continuity of ˆ0 implies ˆ is weakly continuous as well. Let fn be a sequence in GB such that fn Id and ˆ.fn/ A, where A is an 1 ! ! isometry of N . From the previous paragraph, we have that ˆ0.fn/ Id. This implies ! that ˆ .fn/ Id (at least topologically) and therefore ˆ.fn/ Id. In conclusion, A ! ! must be trivial and by Lemma 2.1, the homomorphism ˆ is continuous.

We can now assume that the action of H in N is free. Next, we will prove that if H is inﬁnite, then ˆ is continuous.

Lemma 5.6 If H is inﬁnite and acts freely, then ˆ is weakly continuous.

Proof Let B1 be a ball in M disjoint from B0 . The actions of ˆ.GB1 / and H on the manifold N commute. If we let N N=H , then N is a smooth manifold because 0 WD 0 the action is free. The dimension of N 0 is lower than the dimension of N , moreover, we have that ˆ descends to a homomorphism ˆ GB Diffc.N /. 0W 1 ! 0 By induction, ˆ is continuous. Let p N be an arbitrary point. By Lemma 5.3, 0 2 0 there is a ball B2 B1 such that ˆ .GB /p p. Therefore, ˆ.GB / preserves Â 0 2 D 2 the manifold Hp h.p/ h H , furthermore we can assume that the dimension D f j 2 g of Hp is lower than the dimension of N . In conclusion, we obtain a homomorphism ˆ GB Diffc.Hp/ that by the inductive hypothesis is continuous. 00W 2 ! We are going to deduce the weak continuity of ˆ from the continuity of ˆ0 and ˆ00 . By Lemma 2.1, it is enough to show that if fn GB is a sequence such that fn Id 2 2 ! and ˆ.fn/ A, where A is an isometry of N , then A Id. ! D Take a point q Hp . We are going to show that A.q/ q and that the derivative 2 D Dq.A/ of A at q is trivial. Given that A is an isometry of some metric, these two facts are enough to show that A is trivial. The fact that A.q/ q follows from the D fact that ˆ .fn/ Id. 00 ! We now prove that the derivative DqA is trivial. We can split the tangent space Tq.N / of N at q as Tq.N / Tq.Hp/ W; D ˚

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where Tq.Hp/ is the tangent space of Hp at q and W is the orthogonal complement of Tq.Hp/ in Tq.N /. Observe that as ˆ .fn/ Id, then DqA preserves Tq.Hp/ and 00 ! Dq Id. In a similar way, considering the fact that ˆ .fn/ Id, if we let W jTq .Hp/ D 0 ! denote the orthogonal projection of Tq.N / onto W , we have that W DqA Id. ı D These two facts imply that the linear map Dq.A) must be the identity: Dq.A/ has a matrix decomposition in blocks, each block corresponding to the subspaces Tq.Hp/ and W . In this decomposition, DqA is an upper-triangular matrix and the diagonal blocks are the identity, therefore, all the eigenvalues of DqA are equal to one. Since A is an isometry, DqA is an orthogonal matrix will all its eigenvalues equal to one and therefore DqA must be the identity.

Thus A.q/ q and DqA Id. As A is an isometry, A must be trivial. D D The only remaining case to finish the proof of Lemma 5.4 is the case where H is a finite group and H acts freely on N . We will show that if ˆ were not continuous, we can replace H with an infinite group of isometries H that would reduce the proof of y this case to the previous two lemmas. Lemma 5.7 If H is finite and acts freely, one of the following alternatives hold. (1) ˆ is weakly continuous. (2) There exists a metric on N and an infinite subgroup of isometries H of N , to- n y gether with an embedded ball B R such that the action of ˆ.GB / commutes 0 Â 0 with the action of H . y

Proof Let N0 N , H0 H , N1 N0=H0 and let B1 be a ball in M disjoint WD WD WD from B0 . The homomorphism ˆ descends to a homomorphism ˆ1 GB Diff.N1/ W 1 ! as the groups GB1 and H0 commute with each other. One can check that if ˆ1 is continuous, then ˆ is continuous. Therefore, if ˆ1 is not continuous, there is a sequence fn;1 GB such that fn;1 Id and such that ˆ1.fn;1/ A1 Diff.N1/, 2 1 ! ! 2 where A1 is a nontrivial isometry on some metric on N1 . Let H1 A1 . If H1 does D h i not act freely or H1 is infinite, we can use Lemmas 5.5 and 5.6 to conclude that ˆ1 is continuous; therefore we can assume H1 is finite and fixed point free. We are now in the same situation we were in at the beginning of the proof of this lemma. For that reason, we can repeat this procedure infinitely many times as follows: We construct for each integer k , the following objects: a ball B B disjoint from the k balls Bj for j < k , a closed manifold Nk Nk 1=Hk 1 and a group homomorphism D ˆk GBk Diff.Nk / descending from ˆk 1 . Together with a sequence fn;k GBk W ! f g 2 such that as n , we have that f Id and ˆ .f / A , where A is a ! 1 n;k ! k n;k ! k k nontrivial isometry on some metric on N and the group H A is a nontrivial k k D h k i finite group of diffeomorphisms acting freely on Nk .

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Observe that the only way this procedure cannot be repeated infinitely times is if for some k , the homomorphism ˆk were weakly continuous, but this would imply that ˆ is weakly continuous, therefore we can assume the previous objects are defined for every integer k 1. Next, we are going to show how to get an infinite group of isometries for N from our Hk . Let us define

H h Diffc.N / h is a lift of some diffeomorphism h H : yk D f 2 j k 2 k g Each diffeomorphism hk of Hk can be lifted to a diffeomorphism of N : the diffeomorphism A is the limit of the sequence ˆ .g /. We consider the sequence ˆ.g / k k n;k f n;k g that consist of lifts of ˆk .gn;k /. By Lemma 2.1 this sequence has a convergent subsequence converging to a diffeomorphism Ak0 of N which is necessarily a lift of Ak .

Observe that in fact Hk is a finite subgroup of Diffc.N / and that Hk 1 Hk ; this y y y last inclusion strict. We define the group H S H . Observe that H is necessarily an infinite group. y WD k 0 yk y We will prove that H is invariant under a Riemannian metric on N in a similar way y that we prove Lemma 2.1. In order to do that, we will show the following bounds:

Lemma 5.8 Every h H satisﬁes the bounds 2 y (1) h r Cr for every r 1, k k Ä (2) D.h/ C , k k Ä for the constants C; Cr in Lemma 2.12.

Proof We will prove by induction on k that for every h H , there exists a sequence 2 yk hn GB such that hn Id and ˆ.hn/ h. By Lemma 2.12, this is enough to show 2 ! ! that such bounds in the derivatives hold. This statement is obvious for k 0. Suppose it is true for j k 1. Consider D Ä the projection homomorphism ‰ H H and observe that the kernel of ‰ is W yk ! k exactly Hk 1 . y By definition H A . First, we will show that A has a lift to N belonging to H k D h k i k yk and satisfying our induction hypothesis. Consider the sequence f GB such that n;k 2 k as n , we have that f Id and ˆ .f / A as in the construction of ! 1 n;k ! k n;k ! k the Ak . We have that for every n, the diffeomorphism ˆ.fn;k / is a lift of ˆk .fn;k / to Diffc.N /. Using Lemma 2.1, we can find a subsequence fni ;k of fn;k , such that ˆ.f / A , where A is a diffeomorphism of N . The diffeomorphism A is ni ;k ! k0 k0 k0 necessarily a lift of A and therefore A belongs to H and satisfies our induction k k0 yk hypothesis.

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To finish the induction, observe that as H is cyclic, every element of H is a product k yk of an element of ker.‰/ Hk 1 and a power of Ak0 . Therefore, to conclude that D y every element of H satisfies the induction hypothesis, it is enough to show that if yk f; g H satisfy the induction hypothesis, then fg satisfies the hypothesis too. This 2 yk is easy to prove: if there exist sequences fn ; gn such that fn Id, gn Id and f g f g ! ! such that ˆ.fn/ f , ˆ.gn/ g, then the sequence fngn satisfies that fngn Id ! ! f g ! and ˆ.fngn/ fg. !

To conclude the proof of Lemma 5.7, we would proceed in a similar way as we did in Lemma 2.1. We will show that H preserves a Riemannian metric g in N as follows: y take an arbitrary metric g for N and average g with respect to the groups H . More 0 yn precisely, consider the sequence of metrics 1 X gn h.g0/: D Hn j y j h Hn 2 y Observe that g is invariant under H . Therefore, if there is a convergent subsequence n yn of g converging to a metric g, then g is invariant under H . To ensure the existence n y of such a convergent subsequence, we will use the same argument used in the proof of Lemma 2.1 taking into account the bounds obtained in Lemma 5.8. In conclusion, H y is an inﬁnite group of isometries of N . To ﬁnish the proof of Lemma 5.7, take an embedded ball B B in N disjoint from all the balls Bn and apply Lemma 5.6. 0

This completes the proof of Lemma 5.4.

6 Questions and remarks

6.1 Higher dimensions

Maybe the most natural question to ask in view of Theorem 1.3 is whether it is possible to obtain a characterization of homomorphisms ˆ Diffc.M / Diffc.N / in the case W ! that dim.M / < dim.N /. All the known homomorphisms known to the author are built from pieces, each piece coming from some natural bundles over M or over Symmk .M / (the set of unordered k points in M ). These bundles could be products, coverings, or somewhat similar to the tangent bundle or other bundles where one has some kind of linear action in the ﬁber. The pieces are glued along submanifolds where the actions in both sides agree.

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6.2 Lower regularity

r r One might expect similar results for the groups Diffc .M / of C diffeomorphisms of a manifold M . Some of the techniques here might apply to such problems in the case when r 2 or r 3. One of the reasons to believe it is possible to do so is that Diff .M / Diffr .M / and therefore one might be able to show some sort of c1 c continuity for homomorphisms of the type ˆ Diff .M / Diffr .M / using Militon’s W c1 ! theorem. It is important to point out that Militon’s theorem might be very difﬁcult to generalize to the C r category: one of the main steps in the proof of Militon’s Theorem uses a KAM technique and such techniques typically have a loss of regularity if r < . In the 1 topological category things seem to be much more difﬁcult as every homeomorphim f Homeo0.M / is arbitrarily distorted (see[6]). 2 6.3 Discrete homomorphisms between Lie groups

One might also try to understand discrete homomorphisms between simple Lie groups. For example, one can show that any discrete homomorphism from SO.3/ to itself is conjugate to the standard one. The same is true for SL2.R/. For SL2.C/, the situation is a little bit different, one can use a nontrivial ﬁeld automorphism of C, to get a homomorphism from SL2.C/ to itself that is not conjugate to the standard one. Nonetheless, one can show that any of the homomorphism between SL2.C/ and itself come from this construction. An observation worth mentioning is that if the homomorphism ˆ is measurable, then ˆ is continuous and in fact C 1 ; see Zimmer[21, Appendix B.3]. Therefore, a nonstandard homomorphism is necessarily not measurable (as in the examples for SL2.C/ described above). Other more difﬁcult set of questions related to the Zimmer program (see Fisher[8]) come from asking whether a discrete homomorphism of groups from a simple Lie group G to Diffc.M / comes from a standard embedding. For example, Matsumoto[17] 1 shows that any action of PSL2.R/ in Diffc.S / is conjugate to the standard action. 2 One might wonder if that is also the case for S and PSL3.R/:

2 6.3.1 Question Is every homomorphism ˆ PSL3.R/ Diff.S / conjugate to the W ! standard embedding? 2 Using the fact that PSL2.C/ acts in S and SO.3/ PSL2.C/ one can compose the action of PSL2.C/ by a nontrivial ﬁeld automorphism of C to obtain a nonstandard action of SO.3/ in S2 . Nonetheless one can still ask if the following is true in the volume preserving setting:

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2 6.3.2 Question Does there exist a homomorphism ˆ SO.3/ Diff.S / that is W ! not conjugate to the standard one? For this type of question, the technique used here does not seem to work as the distortion elements in any linear group are at most exponentially distorted. Nevertheless, studying the distorted elements involved might give some useful information.

6.4 Distortion elements in groups of diffeomorphisms

Note that Militon’s theorem implies that if M is a closed manifold and f is an inﬁnite order isometry on M then f is arbitrarily distorted. The converse to this statement is not true: there are examples of diffeomorphisms f in Diffc.M / that are not isometries n but are still recurrent (f is recurrent if it satisﬁes lim infn dC .f ; Id/ 0) and 1 D therefore arbitrarily distorted as a consequence of Militon’s theorem. Some of these examples can be constructed using the Anosov–Katok method. For example, one can use the construction in Herman[13] to obtain such examples.

6.4.1 Question Is it possible to obtain a classiﬁcation of all possible distorted (or 1 2 arbitrarily distorted as deﬁned in[6]) elements in Diff c.S / or Diffc.S /? This might be useful to show that certain discrete groups cannot act by diffeomorphisms on S1 and S2 .

References

[1] J Aramayona, J Souto, Rigidity phenomena in mapping class groups, preprint [2] G Arone, M Kankaanrinta, On the functoriality of the blow-up construction, Bull. Belg. Math. Soc. Simon Stevin 17 (2010) 821–832 MR2777772

[3] A Avila, Distortion elements in Diff1.R=Z/ arXiv:0808.2334 [4] A Banyaga, The structure of classical diffeomorphism groups, Math. and its Appl. 400, Kluwer, Dordrecht (1997) MR1445290 [5] K S Brown, Cohomology of groups, Graduate Texts in Math. 87, Springer, New York (1994) MR1324339 [6] D Calegari, M H Freedman, Distortion in transformation groups, Geom. Topol. 10 (2006) 267–293 MR2207794 [7] R P Filipkiewicz, Isomorphisms between diffeomorphism groups, Ergodic Theory Dynamical Systems 2 (1982) 159–171 MR693972 [8] D Fisher, Groups acting on manifolds: Around the Zimmer program, from: “Geometry, rigidity, and group actions”, (B Farb, D Fisher, editors), Chicago Lectures in Math., Univ. Chicago Press (2011) 72–157 MR2807830

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[9] J Franks, Distortion in groups of circle and surface diffeomorphisms, from: “Dy- namique des difféomorphismes conservatifs des surfaces: un point de vue topologique”, Panor. Synthèses 21, Soc. Math. France, Paris (2006) 35–52 MR2288284 [10] J Franks, M Handel, Distortion elements in group actions on surfaces, Duke Math. J. 131 (2006) 441–468 MR2219247 [11] É Ghys, Prolongements des difféomorphismes de la sphère, Enseign. Math. 37 (1991) 45–59 MR1115743 [12] M Gromov, Asymptotic invariants of inﬁnite groups, from: “Geometric group theory, Volume 2”, (G A Niblo, M A Roller, editors), London Math. Soc. Lecture Note Ser. 182, Cambridge Univ. Press (1993) 1–295 MR1253544 [13] M-R Herman, Construction of some curious diffeomorphisms of the Riemann sphere, J. London Math. Soc. 34 (1986) 375–384 MR856520 [14] M W Hirsch, Differential topology, Graduate Texts in Math. 33, Springer, New York (1976) MR0448362 [15] P Kronheimer, T Mrowka, Monopoles and three-manifolds, New Math. Mono. 10, Cambridge Univ. Press (2007) MR2388043 [16] K Mann, Homomorphisms between diffeomorphism groups, Ergodic Theory Dynam. Systems 35 (2015) 192–214 MR3294298 [17] S Matsumoto, Numerical invariants for semiconjugacy of homeomorphisms of the circle, Proc. Amer. Math. Soc. 98 (1986) 163–168 MR848896 [18] E Militon, Éléments de distorsion du groupe des difféomorphismes isotopes à l’identité d’une variété compacte arXiv:1005.1765 [19] D Montgomery, L Zippin, Topological transformation groups, Interscience, New York (1955) MR0073104 [20] K Parkhe, Smooth gluing of group actions and applications, Proc. Amer. Math. Soc. 143 (2015) 203–212 MR3272745 [21] R J Zimmer, Ergodic theory and semisimple groups, Monographs in Math. 81, Birkhäuser, Basel (1984) MR776417

Department of Mathematics, University of California Berkeley, CA 94720, USA [email protected], [email protected]

Proposed: Danny Calegari Received: 13 September 2013 Seconded: Benson Farb, Leonid Polterovich Revised: 4 August 2014

Geometry & Topology Publications, an imprint of mathematical sciences publishers msp